Smooth models for the Coulomb potential

Abstract

Smooth model potentials with parameters selected to reproduce the spectrum of one-electron atoms are used to approximate the singular Coulomb potential. Even when the potentials do not mimic the Coulomb singularity, much of the spectrum is reproduced within the chemical accuracy. For the Hydrogen atom, the smooth approximations to the Coulomb potential are more accurate for higher angular momentum states. The transferability of the model potentials from an attractive interaction (Hydrogen atom) to a repulsive one (Harmonium and the uniform electron gas) is discussed. I. PHILOSOPHYHow feasible is it to find a model for the Coulomb interaction that is easier to evaluate but still reproduces key properties of the physical interaction? A logical starting point is a system with no interaction, as in KohnSham density functional theory (DFT) [1]. The KohnSham (KS) approximation starts from a non-interacting system, described as the sum of the individual electrons' contributions to the energy:In order to ameliorate the effect of omitting the Coulomb repulsion between the electrons, an extra term, the exchange-correlation functional E xc [ρ], is introduced into the energy expression. For a given external potential v(r) In principle, the KS solutions are exact when E xc is exact, and the KS orbitals yield the exact density of the system with N electrons in the external potential v(r). The accuracy of KS density functional approximations (DFA) depends on the approximation one uses for E xc .The simplest approximation is the local density approximation (LDA) [3][4][5]. In LDA it is assumed that the exchange-correlation functional is local,where the exchange-correlation energy density xc (ρ(r)) at r is taken from the uniform electron gas with density ρ(r).To accurately recover the effect of omitting the interaction between the electrons, one constructs an adiabatic connection that links the KS non-interacting system with the physical interacting system. Traditionally, this adiabatic connection is written as a function of the strength of the interaction, using a simple multiplicative factor λ arXiv:1610.08730v1 [physics.chem-ph]

Correcting Models with Long-Range Electron Interaction Using Generalized Cusp Conditions

“…The interaction between electrons is described by a μ-dependent model potential v int ( r ; μ): μ = 0: there is no interaction between electrons, so v int ( r ; 0) = 0. μ = ∞: we have the physical, Coulomb interaction, so v int ( r ; ∞) = 1/ r . μ∈(0, ∞): we choose v int ( r ; μ ) = w ( r ; μ ) = erf ( μ r ) r where r = r 12 = | r 1 – r 2 |. Exploring other forms of interaction may be both interesting and useful as, for example, in ref . To simplify the notation, we drop μ when μ = ∞.…”

“…μ∈(0, ∞): we choose v int ( r ; μ ) = w ( r ; μ ) = erf ( μ r ) r where r = r 12 = | r 1 – r 2 |. Exploring other forms of interaction may be both interesting and useful as, for example, in ref .…”

“…Furthermore, as u (0) corresponds to the solution at μ = ∞, namely, to the exact solution independent of either λ or μ, we set 0 = . Consequently, the equation for u (1)…”

Interface Dark Excitons at Sharp Lateral Two-Dimensional Heterostructures

“…However, what are the suitable formulas of V I and V C for the numerical calculations? The Coulomb potential and its usage for studying the Hydrogen-like atoms and the dielectric properties of two-dimensional materials have been widely studied [31,[36][37][38][39][40]. Felbacq et al, used the below formula for studying the dielectric properties of two-dimensional materials [40]:…”

A reference‐free stockholder partitioning method based on the force on electrons

“…This is the strategy we explore in the next subsection. Alternatively, one could replace the long‐ranged Coulomb potential with a screened Coulomb potential like the Yukawa potential, the error function potential, or the erfgau interaction …”